## Tag `0791`

Chapter 7: Sites and Sheaves > Section 7.29: Localization of topoi

Lemma 7.29.3. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where $U \in \mathop{\mathrm{Ob}}\nolimits(\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in $\mathcal{C}/\mathcal{F}$ be a family $\{(U_i, s_i) \to (U, s)\}$ such that $\{U_i \to U\}$ is a covering of $\mathcal{C}$. Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous and cocontinuous functor of sites which induces a morphism of topoi $j : \mathop{\mathit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) \to \mathop{\mathit{Sh}}\nolimits(\mathcal{C})$. In fact, there is an equivalence $\mathop{\mathit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) = \mathop{\mathit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal{F}$.

Proof.We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and that $j$ is continuous and cocontinuous. By Lemma 7.20.5 there exists a morphism of topoi $j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$, and there is a left adjoint $j_!$ to $j^{-1}$. A morphism $\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$ is the same thing as a rule which assigns to every pair $(U, s)$ a section $\varphi(s) \in \mathcal{G}(U)$ compatible with restriction maps. Hence this is the same thing as a morphism $\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$. We conclude that $j_!* = \mathcal{F}$. In particular, for every $\mathcal{H} \in \mathop{\mathit{Sh}}\nolimits(\mathcal{C}/\mathcal{F})$ there is a canonical map $$ j_!\mathcal{H} \to j_!* = \mathcal{F} $$ i.e., we obtain a functor $j'_! : \mathop{\mathit{Sh}}\nolimits(\mathcal{C}/\mathcal{F}) \to \mathop{\mathit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$. An inverse to this functor is the rule which assigns to an object $\varphi : \mathcal{G} \to \mathcal{F}$ of $\mathop{\mathit{Sh}}\nolimits(\mathcal{C})/\mathcal{F}$ the sheaf $$ a(\mathcal{G}/\mathcal{F}) : (U, s) \longmapsto \{t \in \mathcal{G}(U) \mid \varphi(t) = s\} $$ We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf and that $a$ is inverse to $j'_!$. $\square$

The code snippet corresponding to this tag is a part of the file `sites.tex` and is located in lines 6571–6583 (see updates for more information).

```
\begin{lemma}
\label{lemma-localize-topos-site}
Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$.
Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where
$U \in \Ob(\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in
$\mathcal{C}/\mathcal{F}$ be a family $\{(U_i, s_i) \to (U, s)\}$
such that $\{U_i \to U\}$ is a covering of $\mathcal{C}$.
Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous
and cocontinuous functor of sites which induces a morphism of topoi
$j : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})$. In fact, there
is an equivalence $\Sh(\mathcal{C}/\mathcal{F}) =
\Sh(\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal{F}$.
\end{lemma}
\begin{proof}
We omit the verification that $\mathcal{C}/\mathcal{F}$ is a site and
that $j$ is continuous and cocontinuous. By
Lemma \ref{lemma-when-shriek} there exists a morphism of topoi
$j$ as indicated, with $j^{-1}\mathcal{G}(U, s) = \mathcal{G}(U)$,
and there is a left adjoint $j_!$ to $j^{-1}$. A morphism
$\varphi : * \to j^{-1}\mathcal{G}$ on $\mathcal{C}/\mathcal{F}$
is the same thing as a rule which assigns to every pair $(U, s)$ a
section $\varphi(s) \in \mathcal{G}(U)$ compatible with restriction maps.
Hence this is the same thing as a morphism
$\varphi : \mathcal{F} \to \mathcal{G}$ over $\mathcal{C}$.
We conclude that $j_!* = \mathcal{F}$. In particular, for every
$\mathcal{H} \in \Sh(\mathcal{C}/\mathcal{F})$ there is a canonical map
$$
j_!\mathcal{H} \to j_!* = \mathcal{F}
$$
i.e., we obtain a functor
$j'_! : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})/\mathcal{F}$.
An inverse to this functor is the rule which assigns to an object
$\varphi : \mathcal{G} \to \mathcal{F}$ of $\Sh(\mathcal{C})/\mathcal{F}$ the
sheaf
$$
a(\mathcal{G}/\mathcal{F}) :
(U, s) \longmapsto \{t \in \mathcal{G}(U) \mid \varphi(t) = s\}
$$
We omit the verification that $a(\mathcal{G}/\mathcal{F})$ is a sheaf
and that $a$ is inverse to $j'_!$.
\end{proof}
```

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